Minimal Injective Resolutions and Auslander-Gorenstein Property for Path Algebras

TL;DR

This paper provides explicit formulas for injective and projective modules in path algebra representations, extends these to minimal injective resolutions, and characterizes when such algebras are Gorenstein.

Contribution

It introduces explicit formulas for injective envelopes and projective covers in path algebra representations and characterizes Gorenstein path algebras based on quiver structure and ring properties.

Findings

Explicit formulas for injective envelopes and projective covers.

Extension of formulas to minimal injective resolutions.

Path algebra is Gorenstein iff the quiver is a line and the ring is Gorenstein.

Abstract

Let $R$ be a ring and $\mathcal{Q}$ be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category $\rm{Rep} (\mathcal{Q} ,R)$ of representations of $\mathcal{Q}$ by left $R$-modules. We also extend our formula to all terms of the minimal injective resolution of $R\mathcal{Q}$. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra $R\mathcal{Q}$ is $k$-Gorenstein if and only if $\mathcal{Q}=\overrightarrow{A_{n}}$ and $R$ is a $k$-Gorenstein ring, where $n$ is the number of vertices of $\mathcal{Q}$.